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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习3.1 - 勒贝格外测度的概念}
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\date{2024 年 4 月 1 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
从日常生活中的长度、面积、体积的概念，可以总结出哪些显然成立的公理？

\vspace{0.2cm}

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\item  %Problem 02
勒贝格的测度概念，包含哪些公理？

\vspace{0.2cm}

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\item  %Problem 3
（以 $n=1,2$ 的情形说明）什么是 $\mathbb{R}^n$ 中的开区间？开区间的体积是怎么定义的？

\vspace{0.2cm}

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\item  %Problem 04
设 $E$ 是 $\mathbb{R}^n$ 的点集。 $E$ 的勒贝格外测度 $m^*(E)$ 是怎么定义的? 

\vspace{0.2cm}

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\item  %Problem 05 H1. 
设 $E\subseteq \mathbb{R}^n$ 是有界点集，证明它的外测度 $m^*(E)<\infty$. 

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\item  %Problem 06 H2. 
证明可数点集的外测度为零。


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\item  %Problem 07
证明外测度具有下述三条基本性质： 
\begin{enumerate}
\item  $m^*(E)\ge 0$, 且当 $E$ 为空集时，$m^*(E)=0$. 
\item  单调性：设 $A\subseteq B$, 则 $m^*(A)\le m^*(B)$. 
\item  次可数可加性：$m^*(\cup_{n=1}^{\infty} A_i) \le \sum_{n=1}^{\infty}m^*(A_i)$. 
\end{enumerate} 
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\item  %Problem 08
设 $E$ 是 $[0,1]$ 中的全体有理数。证明 $m^*(E)=0$.

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\item  %Problem 09
设 $E=(a,b)$ 为直线上的区间。证明 $m^*(E)=b-a$.

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\item  %Problem 10 H3. 
设 $E\subseteq\mathbb{R}$ 是有界点集。%设 $m^*(E)>0$. 
设 $0<c<m^*(E)$. 证明存在子集 $E_1\subseteq E$ 使得 $m^*(E_1)=c$. 

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\end{enumerate}


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\end{document}

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